This study investigates the spatial distribution of inertial particles in turbulent Taylor–Couette flow. Direct numerical simulations are performed using a one-way coupled Eulerian–Lagrangian approach, with a fixed inner-wall Reynolds number of 2500 for the carrier flow, while the particle Stokes number ($St$) varies from 0.034 to 1 for the dispersed phase. We first examine the issue of preferential concentration of particles near the outer-wall region. Employing two-dimensional Voronoï analysis, we observe a pronounced particle clustering with increasing $St$, particularly evident in regions of low fluid velocity. Additionally, we investigate the concentration balance equation, inspired by the work of Johnson et al. (J. Fluid Mech., vol. 883, 2020, A27), to examine the particle radial distribution. We discern the predominant sources of influence, namely biased sampling, turbophoresis and centrifugal effects. Across all cases, centrifugal force emerges as the primary driver, causing particle migration toward the outer wall. Biased sampling predominantly affects smaller inertial particles, driving them toward the inner wall due to sampling within Taylor rolls with inward radial velocity. Conversely, turbophoresis primarily impacts larger inertial particles, inducing migration towards both walls where turbulent intensity is weaker compared with the bulk. With the revealed physics, our work provides a basis for predicting and controlling particle movement and distribution in industrial applications.

We study the melting process of a solid under microgravity, driven solely by lateral vibrations that are perpendicular to the applied temperature gradient due to the absence of gravity-induced convection. Using direct numerical simulations with the phase-field method, we examine two-dimensional vibration-induced melting in a square cavity over four orders of magnitude of vibrational Rayleigh numbers, $10^5\le Ra_{{vib}}\le 10^9$. Our results show that as melting progresses, the flow structure transitions from a periodic-circulation regime with diffusion-dominated heat transfer to a columnar regime with vibroconvection. The mean height of the liquid–solid interface follows a power-law dependency with time, $\bar {\xi } \sim \tilde t^{1/(2-2\alpha )}$, where $\alpha = 0$ in the periodic-circulation regime and $\alpha = 1/2$ in the columnar regime. We further observe that within the columnar regime, the morphological evolution of the liquid–solid interface is influenced by the interaction of columnar thermal plumes in the central regions and the peripheral flow near the sidewalls. Specifically, we offer a comprehensive analysis of the plume merging behaviour, which is governed by the aspect ratio ($\bar {\xi }$) of the liquid layer and the intensity of vibration, quantified by the effective vibrational Rayleigh number $Ra_{vib}^{eff}$. We identify the relationship between the number of columnar plumes $K_m$ and $Ra_{vib}^{eff}$, finding that $K_m \sim \bar {\xi }^{-1} (Ra_{vib}^{eff})^{\gamma }$ with the fitting scaling exponent $\gamma = 0.150 \pm 0.025$. We subsequently quantify the characteristics of the interface roughness amplitude evolution in microgravity vibroconvection. Our results indicate that the roughness amplitude exhibits a power-law dependence on the mean height of the liquid layer. Drawing from the Stefan boundary condition, we theoretically deduce this dependence under the assumption of a non-uniform heat flux distribution at the interface, where the theory is corroborated by our numerical simulations.

Melting and solidification in periodically time-modulated thermal convection are relevant for numerous natural and engineering systems, for example, glacial melting under periodic sun radiation and latent thermal energy storage under periodically pulsating heating. It is highly relevant for the estimation of melt rate and melt efficiency management. However, even the dynamics of a solid–liquid interface shape subjected to a simple sinusoidal heating has not yet been investigated in detail. In this paper, we offer a better understanding of the modulation frequency dependence of the melting and solidification front. We numerically investigate periodic melting and solidification in turbulent convective flow with the solid above and the melted liquid below, and sinusoidal heating at the bottom plate with the mean temperature equal to the melting temperature. We investigate how the periodic heating can prevent the full solidification, and the resulting flow structures and the quasi-equilibrium interface height. We further study the dependence on the heating modulation frequency. As the frequency decreases, we found two distinct regimes, which are ‘partially solid’ and ‘fully solid’. In the fully solid regime, the liquid freezes completely, and the effect of the modulation is limited. In the partially solid regime, the solid partially melts, and a steady or unsteady solid–liquid interface forms depending on the frequency. The interface height can be derived based on the energy balance through the interface. In the partially solid regime, the interface height oscillates periodically, following the frequency of modulation. Here, we propose a perturbation approach that can predict the dependency of the oscillation amplitude on the modulation frequency.

We report the unified constitutive law of vibroconvective turbulence in microgravity, i.e. $Nu \sim a^{-1} Re_{os}^\beta$ where the Nusselt number $Nu$ measures the global heat transport, $a$ is the dimensionless vibration amplitude, $Re_{os}$ is the oscillational Reynolds number and $\beta$ is the universal exponent. We find that the dynamics of boundary layers plays an essential role in vibroconvective heat transport and the $Nu$-scaling exponent $\beta$ is determined by the competition between the thermal boundary layer (TBL) and vibration-induced oscillating boundary layer (OBL). Then a physical model is proposed to explain the change of scaling exponent from $\beta =2$ in the TBL-dominant regime to $\beta = 4/3$ in the OBL-dominant regime. Our finding elucidates the emergence of universal constitutive laws in vibroconvective turbulence, and opens up a new avenue for generating a controllable effective heat transport under microgravity or even microfluidic environment in which the gravity effect is nearly absent.

For dissolving active oil droplets in an ambient liquid, it is generally assumed that the Marangoni effect results in repulsive interactions, while the buoyancy effects caused by the density difference between the droplets, diffusing product and the ambient fluid are usually neglected. However, it has been observed in recent experiments that active droplets can form clusters due to buoyancy-driven convection (Krüger et al., Eur. Phys. J. E, vol. 39, 2016, pp. 1–9). In this study we numerically analyse the buoyancy effect, in addition to the propulsion caused by Marangoni flow (with its strength characterized by the Péclet number $Pe$). The buoyancy effects have their origin in (i) the density difference between the droplet and the ambient liquid, which is characterized by the Galileo number $Ga$; and (ii) the density difference between the diffusing product (i.e. filled micelles) and the ambient liquid, which can be quantified by a solutal Rayleigh number $Ra$. We analyse how the attracting and repulsing behaviour of neighbouring droplets depends on the control parameters $Pe$, $Ga$ and $Ra$. We find that while the Marangoni effect leads to the well-known repulsion between the interacting droplets, the buoyancy effect of the reaction product leads to buoyancy-driven attraction. At sufficiently large $Ra$, even collisions between the droplets can take place. Our study on the effect of $Ga$ further shows that with increasing $Ga$, the collision becomes delayed. Moreover, we derive that the attracting velocity of the droplets, which is characterized by a Reynolds number $Re_d$, is proportional to $Ra^{1/4}/( \ell /R)$, where $\ell /R$ is the distance between the neighbouring droplets normalized by the droplet radius. Finally, we numerically obtain the repulsive velocity of the droplets, characterized by a Reynolds number $Re_{rep}$, which is proportional to $PeRa^{-0.38}$. The balance of attractive and repulsive effect leads to $Pe\sim Ra^{0.63}$, which agrees well with the transition curve between the regimes with and without collision.

We investigate the spatial distribution and dynamics of the vortices in rotating Rayleigh–Bénard convection in a reduced Rayleigh number range $1.3\le Ra/Ra_{c}\le 83.1$. Under slow rotations ($Ra\approx 80\,Ra_{c}$), the vortices are distributed randomly, which is manifested by the size distribution of the Voronoi cells of the vortex centres being a standard $\varGamma$ distribution. The vortices exhibit Brownian-type horizontal motion in the parameter range $Ra\gtrsim 10\,Ra_{c}$. The probability density functions of the vortex displacements are, however, non-Gaussian at short time scales. At modest rotating rates ($4\,Ra_{c}\le Ra\lesssim 10\,Ra_{c}$), the centrifugal force leads to radial vortex motions, i.e. warm cyclones (cold anticyclones) moving towards (outwards from) the rotation axis. The horizontal scale of the vortices decreases with decreasing $Ra/Ra_c$, and the size distribution of their Voronoi cells deviates from the $\varGamma$ distribution. In the rapidly rotating regime ($1.6\,Ra_{c}\le Ra\le 4\,Ra_{c}$), the vortices are densely distributed. The hydrodynamic interaction of neighbouring vortices results in the formation of vortex clusters. Within clusters, cyclones exhibit inverse-centrifugal motion as they submit to the outward motion of the strong anticyclones, and the radial velocity of the anticyclones is enhanced. The radial mobility of isolated vortices, scaled by their vorticity strength, is shown to be a simple power function of the Froude number. For all flow regimes studied, we show that the number of vortices with a lifespan greater than $t$ decreases exponentially as $\exp ({-t/{\tau }})$ for large time, where $\tau$ represents the characteristic lifetime of long-lived vortices.

This study investigates the effect of vibration on the flow structure transitions in thermal vibrational convection (TVC) systems, which occur when a fluid layer with a temperature gradient is excited by vibration. Direct numerical simulation (DNS) of TVC in a two-dimensional enclosed square box is performed over a range of dimensionless vibration amplitudes $0.001 \le a \le 0.3$ and angular frequencies $10^{2} \le \omega \le 10^{7}$, with a fixed Prandtl number of 4.38. The flow visualisation shows the transition behaviour of flow structure upon the varying frequency, characterising three distinct regimes, which are the periodic-circulation regime, columnar regime and columnar-broken regime. Different statistical properties are distinguished from the temperature and velocity fluctuations at the boundary layer and mid-height. Upon transition into the columnar regime, columnar thermal coherent structures are formed, in contrast to the periodic oscillating circulation. These columns are contributed by the merging of thermal plumes near the boundary layer, and the resultant thermal updrafts remain at almost fixed lateral position, leading to a decrease in fluctuations. We further find that the critical point of this transition can be described nicely by the vibrational Rayleigh number ${{Ra}}_{vib}$. As the frequency continues to increase, entering the so-called columnar-broken regime, the columnar structures are broken, and eventually the flow state becomes a large-scale circulation (LSC), characterised by a sudden increase in fluctuations. Finally, a phase diagram is constructed to summarise the flow structure transition over a wide range of vibration amplitude and frequency parameters.

We report that vertical vibration with small amplitude and high frequency can tame convective heat transport in Rayleigh–Bénard convection in a turbulent regime. When vertical vibration is applied, a dynamically averaged ‘anti-gravity’ results that stabilizes the thermal boundary layer and inhibits the eruption of thermal plumes. This eventually leads to the attenuation of the intensity of large-scale mean flow and a significant suppression of turbulent heat transport. Accounting for both the thermally led buoyancy and the vibration-induced anti-gravitational effects, we propose an effective Rayleigh number that helps to extend the Grossmann–Lohse theory to thermal vibrational turbulence. The prediction of the reduction on both the Nusselt and Reynolds numbers obtained by the extended model is found to agree well with the numerical data. In addition, vibrational influences on the mean flow structure and the temporal evolution of Nusselt and Reynolds numbers are investigated. The non-uniform characteristic of vibration-induced ‘anti-gravity’ is discussed. The present findings provide a powerful basis for studying thermal vibrational turbulence and put forward a novel strategy for actively controlling thermal turbulence.

In this paper, we report that reversals of the large-scale circulation in two-dimensional Rayleigh–Bénard (RB) convection can be suppressed by imposing sinusoidally distributed heating to the bottom plate. We examine how the frequency of flow reversals depends on the dimensionless wavenumber $k$ of the spatial temperature modulation with various modulation amplitude $A$. For sufficiently large $k$, the flow reversal frequency is close to that in the standard RB convection under uniform heating. However, when $k$ decreases, the frequency of flow reversal gradually becomes lower and can even be largely reduced. Furthermore, we examine the growth of the corner roll and the global flow structure based on Fourier mode decomposition, and reveal that the size of the corner roll diminishes as the wavenumber decreases. The reason is that the regions occupied by the cold phase can absorb heat from the hot plumes and thus lower their temperature, which reduces the corner roll's kinetic energy input provided by the buoyancy force, and weakens the feeding process of the corner rolls. This results in the locking of the corner roll into a smaller region near the corner, making it harder for a reversal to occur. Using the concept of horizontal convection caused by non-uniform heating, we find a relevant parameter $k/A$ to describe briefly how the reversal frequency depends on wavenumber and modulation amplitude. The present work provides a new way to control the flow reversals in RB convection through modifying temperature boundary conditions.

We numerically investigate turbulent Rayleigh–Bénard convection with gas bubbles attached to the hot plate, mimicking a core feature in electrolysis, catalysis or boiling. The existence of bubbles on the plate reduces the global heat transfer due to the much lower thermal conductivity of gases as compared with liquids and changes the structure of the boundary layers. The numerical simulations are performed in three dimensions at Prandtl number $\mbox{Pr}=4.38$ (water) and Rayleigh number $10^7\leqslant \mbox{Ra}\leqslant 10^8$. For simplicity, we assume the bubbles to be equally sized and having pinned contact lines. We vary the total gas-covered area fraction $0.18 \leqslant S_0 \leqslant 0.62$, the relative bubble height $0.02\leqslant h/H \leqslant 0.05$ (where $H$ is the height of the Rayleigh–Bénard cell), the bubble number $40 \leqslant n \leqslant 144$ and their spatial distribution. In all cases, asymmetric temperature profiles are observed, which we quantitatively explain based on the heat flux conservation at each horizontal section. We further propose the idea of using an equivalent single-phase set-up to mimic the system with attached bubbles. Based on this equivalence, we can calculate the heat transfer. Without introducing any free parameter, the predictions for the Nusselt number, the upper and lower thermal boundary layer thicknesses and the mean centre temperature agree well with the numerical results. Finally, our predictions also work for the cases with much larger $\mbox{Pr}$ (e.g. $400$), which indicates that our results can also be applied to predict the mass transfer in water electrolysis with bubbles attached to the electrode surface or in catalysis.

We numerically investigate turbulent Rayleigh–Bénard convection through two immiscible fluid layers, aiming to understand how the layer thickness and fluid properties affect the heat transfer (characterized by the Nusselt number $\mbox {Nu}$) in two-layer systems. Both two- and three-dimensional simulations are performed at fixed global Rayleigh number $\mbox {Ra}=10^8$, Prandtl number $\mbox {Pr}=4.38$ and Weber number $\mbox {We}=5$. We vary the relative thickness of the upper layer between $0.01 \le \alpha \le 0.99$ and the thermal conductivity coefficient ratio of the two liquids between $0.1 \le \lambda _k \le 10$. Two flow regimes are observed. In the first regime at $0.04\le \alpha \le 0.96$, convective flows appear in both layers and $\mbox {Nu}$ is not sensitive to $\alpha$. In the second regime at $\alpha \le 0.02$ or $\alpha \ge 0.98$, convective flow only exists in the thicker layer, while the thinner one is dominated by pure conduction. In this regime, $\mbox {Nu}$ is sensitive to $\alpha$. To predict $\mbox {Nu}$ in the system in which the two layers are separated by a unique interface, we apply the Grossmann–Lohse theory for both individual layers and impose heat flux conservation at the interface. Without introducing any free parameter, the predictions for $\mbox {Nu}$ and for the temperature at the interface agree well with our numerical results and previous experimental data.

This numerical study presents a simple but extremely effective way to considerably enhance heat transport in turbulent wall-bounded multiphase flows, namely by using oleophilic walls. As a model system, we pick the Rayleigh–Bénard set-up, filled with an oil–water mixture. For oleophilic walls, using only $10\,\%$ volume fraction of oil in water, we observe a remarkable heat transport enhancement of more than $100\,\%$ as compared to the pure water case. In contrast, for oleophobic walls, the enhancement is only of about $20\,\%$ as compared to pure water. The physical explanation of the heat transport increment for oleophilic walls is that thermal plumes detach from the oil-rich boundary layer and carry the heat with them. In the bulk, the oil–water interface prevents the plumes from mixing with the turbulent water bulk and to diffuse their heat. To confirm this physical picture, we show that the minimum amount of oil necessary to achieve the maximum heat transport is set by the volume fraction of the thermal plumes. Our findings provide guidelines of how to optimize heat transport in wall-bounded thermal turbulence. Moreover, the physical insight of how coherent structures are coupled with one of the phases of a two-phase system has very general applicability for controlling transport properties in other turbulent wall-bounded multiphase flows.

Indoor ventilation is essential for a healthy and comfortable living environment. A key issue is to discharge anthropogenic air contamination such as CO$_2$ gas or, of potentially more direct consequence, airborne respiratory droplets. Here, by employing direct numerical simulations, we study mechanical displacement ventilation with a wide range of ventilation rates $Q$ from 0.01 to 0.1 m$^3$ s$^{-1}$ person$^{-1}$. For this ventilation scheme, a cool lower zone is established beneath a warm upper zone with interface height $h$, which depends on $Q$. For weak ventilation, we find the scaling relation $h\sim Q^{3/5}$, as suggested by Hunt & Linden (Build. Environ., vol. 34, 1999, pp. 707–720). Also, the CO$_{2}$ concentration decreases with $Q$ within this regime. However, for too strong ventilation, the interface height $h$ becomes insensitive to $Q$, and the ambient averaged CO$_2$ concentration decreases towards the ambient value. At these values of $Q$, the concentrations of pollutants are very low and so further dilution has little effect. We suggest that such scenarios arise when the vertical kinetic energy associated with the ventilation flow is significant compared with the potential energy of the thermal stratification.

We theoretically and numerically investigate the instabilities driven by diffusiophoretic flow, caused by a solutal concentration gradient along a reacting surface. The important control parameters are the Péclet number $Pe$, which quantifies the ratio of the solutal advection rate to the diffusion rate, and the Schmidt number $Sc$, which is the ratio of viscosity and diffusivity. First, we study the diffusiophoretic flow on a catalytic plane in two dimensions. From a linear stability analysis, we obtain that for $Pe$ larger than $8{\rm \pi}$ mass transport by convection overtakes that by diffusion, and a symmetry-breaking mode arises, which is consistent with numerical results. For even larger $Pe$, nonlinear terms become important. For $Pe > 16{\rm \pi}$, multiple concentration plumes are emitted from the catalytic plane, which eventually merge into a single larger one. When $Pe$ is even larger ($Pe \gtrsim 603$ for Schmidt number $Sc=1$), there are continuous emissions and merging events of the concentration plumes. The newly found flow states have different flow structures for different $Sc$: for $Sc\geqslant 1$, we observe the chaotic emission of plumes, but the fluctuations of concentration are only present in the region near the catalytic plane. In contrast, for $Sc<1$, chaotic flow motion occurs also in the bulk. In the second part of the paper, we conduct three-dimensional simulations for spherical catalytic particles, and beyond a critical Péclet number again find continuous plume emission and plume merging, now leading to a chaotic motion of the phoretic particle. Our results thus help us to understand the experimentally observed chaotic motion of catalytic particles in the high $Pe$ regime.

It is commonly accepted that the breakup criteria of drops or bubbles in turbulence is governed by surface tension and inertia. However, also buoyancy can play an important role at breakup. In order to better understand this role, here we numerically study two-dimensional Rayleigh–Bénard convection for two immiscible fluid layers, in order to identify the effects of buoyancy on interface breakup. We explore the parameter space spanned by the Weber number $5\leqslant We \leqslant 5000$ (the ratio of inertia to surface tension) and the density ratio between the two fluids $0.001 \leqslant \varLambda \leqslant 1$, at fixed Rayleigh number $Ra=10^8$ and Prandtl number $Pr=1$. At low $We$, the interface undulates due to plumes. When $We$ is larger than a critical value, the interface eventually breaks up. Depending on $\varLambda$, two breakup types are observed. The first type occurs at small $\varLambda \ll 1$ (e.g. air–water systems) when local filament thicknesses exceed the Hinze length scale. The second, strikingly different, type occurs at large $\varLambda$ with roughly $0.5 < \varLambda \leqslant 1$ (e.g. oil–water systems): the layers undergo a periodic overturning caused by buoyancy overwhelming surface tension. For both types, the breakup criteria can be derived from force balance arguments and show good agreement with the numerical results.

Micro- and nanodroplets have many important applications such as in drug delivery, liquid–liquid extraction, nanomaterial synthesis and cosmetics. A commonly used method to generate a large number of micro- or nanodroplets in one simple step is solvent exchange (also called nanoprecipitation), in which a good solvent of the droplet phase is displaced by a poor one, generating an oversaturation pulse that leads to droplet nucleation. Despite its crucial importance, the droplet growth resulting from the oversaturation pulse in this ternary system is still poorly understood. We experimentally and theoretically study this growth in Hele-Shaw-like channels by measuring the total volume of the oil droplets that nucleate out of it. In order to prevent the oversaturated oil from exiting the channel, we decorated some of the channels with a porous region in the middle. Solvent exchange is performed with various solution compositions, flow rates and channel geometries, and the measured droplets volume is found to increase with the Péclet number, $Pe$, with an approximate effective power law $V\propto Pe^{0.50}$. A theoretical model is developed to account for this finding. With this model we can indeed explain the $V\propto Pe^{1/2}$ scaling, including the prefactor, which can collapse all data of the ‘porous’ channels onto one universal curve, irrespective of channel geometry and composition of the mixtures. Our work provides a macroscopic approach to this bottom-up method of droplet generation and may guide further studies on oversaturation and nucleation in ternary systems.

Rayleigh–Bénard (RB) convection with free-slip plates and horizontally periodic boundary conditions is investigated using direct numerical simulations. Two configurations are considered, one is two-dimensional (2-D) RB convection and the other one three-dimensional (3-D) RB convection with a rotating axis parallel to the plate, which for strong rotation mimics 2-D RB convection. For the 2-D simulations, we explore the parameter range of Rayleigh numbers $Ra$ from $10^{7}$ to $10^{9}$ and Prandtl numbers $Pr$ from $1$ to $100$. The effect of the width-to-height aspect ratio $\varGamma$ is investigated for $1\leqslant \varGamma \leqslant 128$. We show that zonal flow, which was observed, for example, by Goluskin et al. (J. Fluid. Mech., vol. 759, 2014, pp. 360–385) for $\varGamma =2$, is only stable when $\varGamma$ is smaller than a critical value, which depends on $Ra$ and $Pr$. The regime in which only zonal flow can exist is called the first regime in this study. With increasing $\varGamma$, we find a second regime in which both zonal flow and different convection roll states can be statistically stable. For even larger $\varGamma$, in a third regime, only convection roll states are statistically stable and zonal flow is not sustained. How many convection rolls form (or in other words, what the mean aspect ratio of an individual roll is), depends on the initial conditions and on $Ra$ and $Pr$. For instance, for $Ra=10^{8}$ and $Pr=10$, the aspect ratio $\varGamma _r$ of an individual, statistically stable convection roll can vary in a large range between $16/11$ and $64$. A convection roll with a large aspect ratio of $\varGamma _r = 64$, or more generally already with $\varGamma _r \gg 10$, can be seen as ‘localized’ zonal flow, and indeed carries over various properties of the global zonal flow. For the 3-D simulations, we fix $Ra=10^{7}$ and $Pr=0.71$, and compare the flow for $\varGamma =8$ and $\varGamma = 16$. We first show that with increasing rotation rate both the flow structures and global quantities like the Nusselt number $Nu$ and the Reynolds number $Re$ increasingly behave like in the 2-D case. We then demonstrate that with increasing aspect ratio $\varGamma$, zonal flow, which was observed for small $\varGamma =2{\rm \pi}$ by von Hardenberg et al. (Phys. Rev. Lett., vol. 15, 2015, 134501), completely disappears for $\varGamma =16$. For such large $\varGamma$, only convection roll states are statistically stable. In-between, here for medium aspect ratio $\varGamma = 8$, the convection roll state and the zonal flow state are both statistically stable. What state is taken depends on the initial conditions, similarly as we found for the 2-D case.

In the preparation of café latte, spectacular layer formation can occur between the espresso shot in a glass of milk and the milk itself. Xue et al. (Nat. Commun., vol. 8, 2017, pp. 1–6) showed that the injection velocity of espresso determines the depth of coffee–milk mixture. After a while, when a stable stratification forms in the mixture, the layering process can be modelled as a double diffusive convection system with a stably stratified coffee–milk mixture cooled from the side. More specifically, we perform (two-dimensional) direct numerical simulations of laterally cooled double diffusive convection for a wide parameter range, where the convective flow is driven by a lateral temperature gradient while stabilized by a vertical concentration gradient. Depending on the strength of stabilization as compared to the thermal driving, the system exhibits different flow regimes. When the thermal driving force dominates over the stabilizing force, the flow behaves like vertical convection in which a large-scale circulation develops. However, with increasing strength of the stabilizing force, a meta-stable layered regime emerges. Initially, several vertically-stacked convection rolls develop, and these well-mixed layers are separated by sharp interfaces with large concentration gradients. The initial thickness of these emerging layers can be estimated by balancing the work exerted by thermal driving and the required potential energy to bring fluid out of its equilibrium position in the stably stratified fluid. In the layered regime, we further observe successive layer merging, and eventually only a single convection roll remains. We elucidate the following merging mechanism: as weakened circulation leads to accumulation of hot fluid adjacent to the hot sidewall, larger buoyancy forces associated with hotter fluid eventually break the layer interface. Then two layers merge into a larger layer, and circulation establishes again within the merged structure.

We perform a numerical study of the heat transfer and flow structure of Rayleigh–Bénard (RB) convection in (in most cases regular) porous media, which are comprised of circular, solid obstacles located on a square lattice. This study is focused on the role of porosity $\unicode[STIX]{x1D719}$ in the flow properties during the transition process from the traditional RB convection with $\unicode[STIX]{x1D719}=1$ (so no obstacles included) to Darcy-type porous-media convection with $\unicode[STIX]{x1D719}$ approaching 0. Simulations are carried out in a cell with unity aspect ratio, for Rayleigh number $Ra$ from $10^{5}$ to $10^{10}$ and varying porosities $\unicode[STIX]{x1D719}$, at a fixed Prandtl number $Pr=4.3$, and we restrict ourselves to the two-dimensional case. For fixed $Ra$, the Nusselt number $Nu$ is found to vary non-monotonically as a function of $\unicode[STIX]{x1D719}$; namely, with decreasing $\unicode[STIX]{x1D719}$, it first increases, before it decreases for $\unicode[STIX]{x1D719}$ approaching 0. The non-monotonic behaviour of $Nu(\unicode[STIX]{x1D719})$ originates from two competing effects of the porous structure on the heat transfer. On the one hand, the flow coherence is enhanced in the porous media, which is beneficial for the heat transfer. On the other hand, the convection is slowed down by the enhanced resistance due to the porous structure, leading to heat transfer reduction. For fixed $\unicode[STIX]{x1D719}$, depending on $Ra$, two different heat transfer regimes are identified, with different effective power-law behaviours of $Nu$ versus $Ra$, namely a steep one for low $Ra$ when viscosity dominates, and the standard classical one for large $Ra$. The scaling crossover occurs when the thermal boundary layer thickness and the pore scale are comparable. The influences of the porous structure on the temperature and velocity fluctuations, convective heat flux and energy dissipation rates are analysed, further demonstrating the competing effects of the porous structure to enhance or reduce the heat transfer.

We numerically investigate both single and multiple droplet dissolution with droplets consisting of less dense liquid dissolving in a denser host liquid. In this situation, buoyancy can lead to convection and thus plays an important role in the dissolution process. The significance of buoyancy is quantified by the Rayleigh number $Ra$, which is the buoyancy force over the viscous damping force. In this study, $Ra$ spans almost four decades from 0.1 to 400. We focus on how the mass flux, characterized by the Sherwood number $Sh$, and the flow morphologies depend on $Ra$. For single droplet dissolution, we first show the transition of the $Sh(Ra)$ scaling from a constant value to $Sh\sim Ra^{1/4}$, which confirms the experimental results by Dietrich et al. (J. Fluid Mech., vol. 794, 2016, pp. 45–67). The two distinct regimes, namely the diffusively and the convectively dominated regimes, exhibit different flow morphologies: when $Ra\geqslant 10$, a buoyant plume is clearly visible, which contrasts sharply with the pure diffusion case at low $Ra$. For multiple droplet dissolution, the well-known shielding effect comes into play at low $Ra$, so that the dissolution rate is slower as compared to the single droplet case. However, at high $Ra$, convection becomes more and more dominant so that a collective plume enhances the mass flux, and remarkably the multiple droplets dissolve faster than a single droplet. This has also been found in the experiments by Laghezza et al. (Soft Matt., vol. 12 (26), 2016, pp. 5787–5796). We explain this enhancement by the formation of a single, larger plume rather than several individual plumes. Moreover, there is an optimal $Ra$ at which the enhancement is maximized, because the single plume is narrower at larger $Ra$, which thus hinders the enhancement. Our findings demonstrate a new mechanism in collective droplet dissolution, which is the merging of the plumes, which leads to non-trivial phenomena, contrasting the shielding effect.